The paper studies two formal schemes related to ω-completeness.
LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where α(x) is a formula with the only free variable x): (a) (for anyn) (⊢ T α(n)), (b) ⊢ T ∀ x Pr T (−α(x)−) and (c) ⊢ T ∀xα(x) (the notational conventions are those of Smoryński ). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) ⇒ (c) and (a) ⇒ (b), where α ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) ⇒ (c) is equivalent to ¬ConsT and 2. the schema corresponding to (a) ⇒ (b) is not consistent with 1-CONT. The former result follows from a simple adaptation of the ω-incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma.
KeywordsMathematical Logic Formal Theory Base theoryS Computational Linguistic Formal Extension
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